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Chapter 11: Problem 10

Does a particle moving at constant speed in a straight line have angular momentum about a point on the line? About a point not on the line? In either case, is its angular momentum constant?

Short Answer

Expert verified
The angular momentum of a particle moving at constant speed in a straight line is zero about a point on the line and non-zero but constant about a point not on the line.

Step by step solution

01

Understand Angular Momentum

Angular momentum, denoted as \( L \), of a particle is defined by \( L = mv \times r \), where \( m \) is the mass of the particle, \( v \) is its velocity, and \( r \) is the position vector of the particle with respect to the point about which the angular momentum is to be measured. Here, \( v \times r \) denotes the cross product of the velocity and position vectors, which yields a vector that is perpendicular to the plane formed by the two input vectors.
02

Determine Angular Momentum on the Line

In the context of the point being on the line of motion, the particle's velocity vector and position vector lie along the same line, implying that the angle between them is zero. As the cross product of two vectors is zero if the vectors are either identical or opposite, the angular momentum for this case is zero (since \( v \times r = 0 \)). This value doesn't change as the particle continues to move at a constant speed along the straight line.
03

Determine Angular Momentum not on the Line

In the context of the point not being on the line of motion, the particle's velocity vector and position vector do not coincide. This means that the cross product \( v \times r \) isn't zero. Hence, the particle has some angular momentum relative to this point. As the particle keeps moving with constant speed, the distance \( r \) is changing but in a way that the absolute value of the cross product remains constant by keeping a constant angle. Hence, the angular momentum remains constant for this case.

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Most popular questions from this chapter

A car is headed north at \(70 \mathrm{km} / \mathrm{h}\). Give the magnitude and direction of the angular velocity of its 62 -cm-diameter wheels.

Pulsars- -the rapidly rotating neutron stars described in Example 11.2 - have magnetic fields that interact with charged particles in the surrounding interstellar medium. The result is torque that causes the pulsar's spin rate and therefore its angular momentum to decrease very slowly. The table below gives values for the rotation period of a given pulsar as it's been observed at the same date every 5 years for two decades. The pulsar's rotational inertia is known to be \(1.12 \times 10^{38} \mathrm{kg} \cdot \mathrm{m}^{2} .\) Make a plot of the pulsar's angular momentum over time, and use the associated best-fit line, along with the rotational analog of Newton's law, to find the torque acting on the pulsar. $$\begin{array}{|l|c|c|c|c|c|}\hline \text { Year of observation } & 1995 & 2000 & 2005 & 2010 & 2015 \\\\\hline \begin{array}{l} \text { Angular momentum } \\\\\left(10^{37} \mathrm{kg} \cdot \mathrm{m}^{2} / \mathrm{s}\right)\end{array} & 7.844 & 7.831 & 7.816 & 7.799 & 7.787 \\\\\hline\end{array}$$

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Express the units of angular momentum (a) using only the fundamental units kilogram, meter, and second; (b) in a form involving newtons; (c) in a form involving joules.
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